In mathematics, the Genocchi numbers G<sub>n</sub>, named after Angelo Genocchi, are a sequence of integers that satisfy the relation
The first few Genocchi numbers are 0, 1, −1, 0, 1, 0, −3, 0, 17 , see .
Properties
- The generating function definition of the Genocchi numbers implies that they are rational numbers. In fact, G<sub>2n+1</sub> = 0 for n ≥ 1 and (−1)<sup>n</sup>G<sub>2n</sub> is an odd positive integer.
- Genocchi numbers G<sub>n</sub> are related to Bernoulli numbers B<sub>n</sub> by the formula
Combinatorial interpretations
The exponential generating function for the signed even Genocchi numbers (−1)<sup>n</sup>G<sub>2n</sub> is
They enumerate the following objects:
- Permutations in S<sub>2n−1</sub> with descents after the even numbers and ascents after the odd numbers.
- Permutations π in S<sub>2n−2</sub> with 1 ≤ π(2i−1) ≤ 2n−2i and 2n−2i ≤ π(2i) ≤ 2n−2.
- Pairs (a<sub>1</sub>,...,a<sub>n−1</sub>) and (b<sub>1</sub>,...,b<sub>n−1</sub>) such that a<sub>i</sub> and b<sub>i</sub> are between 1 and i and every k between 1 and n−1 occurs at least once among the a<sub>i</sub>'s and b<sub>i</sub>'s.
- Reverse alternating permutations a<sub>1</sub> < a<sub>2</sub> > a<sub>3</sub> < a<sub>4</sub> >...>a<sub>2n−1</sub> of [2n−1] whose inversion table has only even entries.
Primes
The only known prime numbers which occur in the Genocchi sequence are 17, at n = 8, and âÂÂ3, at n = 6 (depending on how primes are defined). It has been proven that no other primes occur in the sequence
See also
References